Rotor Systems: Analysis and Identification 1st Edition Rajiv Tiwari
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Identification of Continuous-Time Systems: Linear and Robust Parameter Estimation (Engineering Systems and Sustainability) 1st Edition Allamaraju Subrahmanyam
Indian Institute of Technology Guwahati, Guwahati, India
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Library of Congress Cataloging-in-Publication Data
5.2 Gyroscopic Moments in Rotating Systems ................................................................................191
5.2.1 Motion of a Rotor Mounted on Two Bearings ...........................................................191
5.2.2 Gyroscopic Moments through the Coriolis Component of Accelerations............................................................................................................... 193
5.2.3 Gyroscopic Moments in a Rotating Thin Blade .........................................................194
5.2.4 Gyroscopic Moments in a Multibladed Propeller .....................................................196
5.3 Synchronous Motion of Rotors....................................................................................................198
5.3.1 A Cantilever Rotor with a Thin Disc ...........................................................................198
5.3.2 A Cantilever Rotor with a Long Stick ........................................................................203
5.4 Asynchronous Rotational Motion of Rotor System ................................................................208
5.5 Asynchronous General Motion of Rotor Systems ....................................................................213
5.6 Gyroscopic Effects by the Dynamics Approach ......................................................................227
5.7 Analysis of Gyroscopic Effects with Energy Methods .............................................................231
5.8 Pure Transverse Rotational Vibrations of a Jeffcott Rotor Model with Moment Unbalance ................................................................................................237
6.6 Simple Geared Rotor Systems .....................................................................................................297
6.7 TMM for Branched Geared Rotor Systems ..............................................................................306
6.8 TMM for Damped Torsional Vibrations ...................................................................................314
6.9 Modeling of Reciprocating Machine Systems ...........................................................................318
6.9.1 An Equivalent Polar Mass Moment of Inertia ...........................................................318
6.9.1.1 Due to Revolving and Reciprocating Masses of the Connecting Rod ............................................................................................318
6.9.1.2 Due to Revolving Masses on the Crank ....................................................319
6.9.1.3 Due to Reciprocating Masses of the Piston ..............................................319
6.9.2 Equivalent Torsional Stiffness of Crankshafts ...........................................................321
6.9.3 Torque Variations in Reciprocating Machinery .......................................................322
Appendix 9.1 Comparison of Transverse Vibration continuous and finite element Analyses in y-z plane with that of z-x plane ..............................................................................545 References..................................................................................................................................................547
Transverse Vibrations of Rotor Systems with Higher-Order Effects by the Continuous and Finite Element Methods
10.1 Gyroscopic Effects in Rotor Systems with a Single Rigid Disc ..............................................549
10.2 Timoshenko Beam Theory ..........................................................................................................562
10.3 Finite-Element Formulations of the Timoshenko Beam ........................................................569
10.3.1 Weak Formulations of the Timoshenko Beam Element for the Static Case ................................................................................................................570
10.3.2 Derivation of Shape Functions .....................................................................................571
10.3.3 Weak Formulation of the Timoshenko Beam Element for the Dynamic Case .........................................................................................................577
10.4 Whirling of Timoshenko Shafts .................................................................................................583
10.4.1 Equations of Motion of a Spinning Timoshenko Shaft ...........................................584
10.4.3 The Weak Form Finite-Element Formulations .........................................................589
10.4.4 Rigid Disc Element ........................................................................................................590
10.4.5 System Equations of Motion ........................................................................................590 10.4.6 Eigenvalue Problem.......................................................................................................590 10.5 Concluding Remarks....................................................................................................................594 Exercise Problems
ko
11 Instability Analysis of Simple Rotor Systems
11.1 Self-Excited Vibrations ................................................................................................................605 11.2 Phenomenon of the Oil Whirl ....................................................................................................608
Instability
11.5 Phenomenon of the Oil Whip......................................................................................................617
11.6 Instability Analysis due to Internal Damping in Rotors .........................................................619
11.7 Instability Due to Rotor Polar Asymmetry ..............................................................................629
11.8 Instability of An Asymmetric Continuous Rotor ...................................................................634
11.8.1 Equations of Motion .....................................................................................................634
11.8.2 Support Conditions and Characteristic Equation....................................................640
11.8.3 Whirl Natural Frequencies and Critical Speeds .......................................................643
11.8.4 Stability Analysis of Asymmetric Shaft with Gyroscopic Effects ..........................646
11.9 Rotor System with Variable Stiffness Characteristics .............................................................648
11.9.1 A Rotor System with Variable Stiffness .....................................................................650
11.9.2 Physical Analysis of a Horizontal Asymmetric Shaft with Gravity Effects ................................................................................................................
11.9.3 Analytical Solution of the Equation of Motion of Asymmetric Rotor .................653
11.10 Subcritical Vibrations of a Jeffcott Rotor ..................................................................................659
11.11 Instability Analysis Due to Stream Whirl ................................................................................ 664
11.12 Instability Analysis Due to Rotary Seals...................................................................................668
11.13 Analysis of Nonlinear Equations of Motion of the Jeffcott Rotor (Run-up and Rundown) ......................................................................................................................................670 11.14 Concluding Remarks....................................................................................................................675
Problems
12 Instability of Flexible Rotors Mounted on Flexible Bearings
12.1 Flexible Rotors Mounted on Flexible Bearings ........................................................................685
12.1.1 Fluid-Film
12.1.2
13.1 Unbalances in Rigid and Flexible Rotors ..................................................................................
13.1.1 Unbalance in a Single Plane .........................................................................................708
13.1.2 Unbalances in Two or More Planes ............................................................................709
13.2 Principles of Rigid Rotor Balancing ...........................................................................................710
13.4.2.2 Basic Theory of Modal Balancing .............................................................733
13.4.2.3 Illustration of Modal Balancing Method up to the Second Mode ................................................................................................735
15.5.4.1 Determining Natural Frequencies of the Rotor Bearing System Using Impact-Hammer Test.........................................................................851
16.1 Visual Presentation of Vibration Measurements......................................................................861
16.2 Errors in Vibrat ion Acquisitions ................................................................................................866
16.3 Basic Concepts of Fourier Series ................................................................................................873
16.3.1 Real Fourier Series.........................................................................................................874
16.3.2 Complex Fourir Series .................................................................................................. 874
16.4 Basics of Fourier Transform and Fourier Integral ..................................................................878
16.5 Basics of the Discrete Fourier Transform ................................................................................. 880
16.6 Basics of the Fast Fourier Transform.........................................................................................887
16.7 Leakage Error and Its Remediation ...........................................................................................888
16.7.1 Remediation of Leakage Errors by the Windowing Function ...............................890
16.7.2 Avoidance of Leakage by Coinciding Periods...........................................................892
16.8 Full Spectrum and Its Applications to Rotor Dynamic Analysis .........................................893
16.8.1 Full Spectrum from Orbit Plots ..................................................................................894
16.8.2 Full Spectrum from Half Spectrum ...........................................................................896
16.8.3 Full Spectrum from Real DFT (or Half Spectrum) ..................................................900
16.8.4 Full Spectrum from Complex DFT ............................................................................902
16.8.5 Phase Ambiguity in the Full Spectrum .....................................................................904
16.8.6 Multiharmonic Quadrature Reference Signal and Phase Compensation Algorithm ............................................................................................907
16.9 Statistical Properties of Random Discrete Signals ..................................................................908
16.9.1 Probability, Probability Distribution Function, and Probability Density Function ...........................................................................................................908
16.9.2 Random Process, Ensemble, and Sample Function ..................................................912
16.9.3 Stationary and Ergodic Processes ................................................................................913
16.9.4 Estimation of Probability Distribution and Probability Distribution Function .....914
18.4 Block Diagrams and Transfer Functions .................................................................................1005
18.4.1 Block Diagrams and Transfer Functions of Magnetic Bearing Systems .............1005
18.4.2 The PID Controller and Its Transfer Function ........................................................1007
18.4.3 Transfer Function and Block Diagram of the Overall Active Magnetic Bearing System..............................................................................................................1007
18.5 Tuning of the Controller Parameters ...................................................................................... 1009
This book is the outcome of an elective course on “Rotor Dynamics” offered by me to undergraduate, graduate, and postgraduate students at IIT Guwahati over the last 18 years. It contains material on some of the research work done by me with my graduate students. Hopefully, the content will be useful for classroom teaching and will serve as a reference book for pursuing research and development in the field of rotor dynamics.
In a broader sense, rotor dynamics covers several topics, namely modeling, analysis, measurement, signal processing, identification, condition monitoring, and control of rotor-bearing systems. The modeling and analysis of rotor-bearing dynamics have now reached a mature state. The finite element method (FEM) and the transfer matrix method (TMM) have been used extensively for modeling and analyses of rotor systems. Until today, monitoring the condition of rotor-bearing systems based on vibrations was mainly concerned with feature-based fault detection and diagnostics. As a result of this, the methods available so far are not reliable and failsafe according to the expectations of fellow engineers working in the field. For model-based condition monitoring of rotor-bearing systems, identification methods for system parameters are under development. In terms of identifying rotor system parameters, from the available literature a lot of possibilities have appeared in the field. The purpose of this course material is to give a basic understanding of the rotor dynamics phenomena with the help of a few simple rotor models and modern analysis methods for real-life rotor systems. This background will be helpful in the identification of rotor-bearing system parameters and its use in futuristic model-based condition monitoring and fault diagnosis and prognostics.
The content of the present book is in two major parts as title of the book also suggests, the first is the modeling and analysis of rotor phenomena, and the second is the condition monitoring of rotor systems and associated identification of system parameters. The first part basically deals with theory related to dynamic analysis of simple rotors, which will give a systematic presentation of some of the rotor dynamic phenomena in the transverse and torsional vibrations with the help of simple rotor mathematical models. But for analyzing complex rotors that are used in practice, the free and forced response analyses of multi-DOF rotor system have been presented. The Euler-Bernoulli and Timoshenko beam models have been considered in transverse vibration analysis. The present book material keeps a balance between the transfer matrix method (TMM) and the finite element method (FEM) for the analysis of complex rotors. However, the FEM is the most practical approach and it can be applied to very large complex rotor-bearing-foundation systems with its easy implementation in computer code and due to development in the condensation or reduction schemes to reduce computation effort and time. The dynamic analysis of rotors contains finding the whirl natural frequency, mode shapes, Campbell diagram, critical speeds, unbalance responses and instability frequency bands. Apart from conventional bearings, i.e. the rolling element and hydrodynamic bearings, and dynamic seals and dampers, the present book also touch upon the contemporary active magnetic bearings and associated rotor unbalance response and instability analyses for the rigid and flexible rotors. The second part basically contains the practical aspect of rotor systems related to the condition monitoring and system identification.
Vibration measurements commonly used in rotor systems has been covered in the book. Apart from that the signal processing of measured vibrations and its display in various forms have been given in great details especially for rotor system applications. Simple frequency domain signal processing techniques, and associated error involved and corrective methods to be taken have been presented. Apart from this, specialized frequency domain signal processing in the form of full spectrum to show the forward and backward whirling components in vibration signals from a rotor system, is one such kind of signal processing given in detail. Some of the basic statistical properties of random measurement signals are provided, which is useful for feature based condition monitoring of rotating machinery. The rigid and flexible rotor balancing, and the experimental identification of bearing dynamic parameters are covered in detail. Condition monitoring of simple machine elements and subsystems based on time and frequency domain signals have been presented. It covers, unbalance, shaft bow, rubs, shaft cracks, shaft misalignment, rolling element bearings, gears, pumps and induction motors.
Extensive information is available on feature based identification different types of fault in the rotor systems. These methods are being used by industries. System parameters identification algorithms are being developed which are used to characterize critical component parameters with the help of experimental measurements and to identify the fault. However, still they have not been used in practice. The present book material compiles some of the available literature in a systematic and lucid form in respective chapters so as to boost research in the developing area of the rotor dynamics.
The book is supplemented throughout by both manual and computer-based calculations. It is expected that with this book, students will receive sufficient exposure and motivation to apply the finite element method and transfer matrix method to rotor dynamics and allied areas. Exercise problems would definitely enlarge the understanding of the subject and some of them can be taken as term projects due to its effort involvement.
As the title suggests, Chapters 1 to 12 and Chapter 18 are on the dynamic analysis of rotor systems, whereas Chapters 13 to 17 are on the rotor system identification and its condition monitoring. Chapter 1 gives a historical development of the subject through problems faced by fellow engineers in the field in addition to the overall progress that has been made. In terms of analysis, a major portion of the text is devoted to finding the rotor system’s natural frequencies (free vibration) and critical speeds (forced vibration) for transverse vibrations (Chapters 2 to 5, 8 to 10, and 12) as well as torsional vibration (Chapters 6 and 7). Chapter 3 provides analyses of various bearings, seals, and dampers, as well as how to obtain rotordynamic parameters for them to be used for rotordynamic analysis. Chapters 11 and 12 are devoted to the analysis of instabilities in rotor systems due to various sources. Dynamic balancing of rotors is covered in Chapter 13, and Chapter 14 provides the experimental methodology of identifying rotordynamic parameters for bearings, seals, and dampers. Chapter 15 describes various instruments used for rotordynamic measurement purposes, and Chapter 16 describes the processing of measured signals to be used for rotordynamic identification. Monitoring the condition of various rotor elements and systems is covered in Chapter 17. Analysis of rotors integrated with active magnetic bearings is covered in Chapter 18, with an emphasis on dynamics and control of the rigid and flexible rotors. Overall the book contains all the facets of the rotor analysis and identification. The content of the book can be judiciously chosen for a semester course depending upon the focus of the course. For example, Chapters 1 through 9 and 11 through 13 can be chosen for 40 hours lectures, except more details of Chapter 3 can be excluded. However, if the TMM is to be excluded from Chapters 6 and 8, then selected topics from remaining chapters can be taken, especially from Chapter 18.
I sincerely acknowledge the Ministry of Human Resources and Development, New Delhi, for funding towards the development of the web and video courses under the National Programme on Technology Enhanced Learning (NPTEL) on the subject. I would like to express special thanks and gratitude to my teachers (Dr N. S. Vyas, Dr B. P. Singh, Dr K. Gupta, Dr J. S. Rao, Dr A. W. Lees, Dr M. I. Friswell, and Dr R. Markert who are well-known figures in the field of rotor dynamics) and academic collaborators friends (Dr J. K. Sinha, Dr Fadi Dohnal Dr S. Jana, Dr A. S. Sekhar, Dr A. Darpe, Dr M. Tiwari, Dr A. A. Khan and Dr A. Chasalevris). My heartfelt thanks to the help offered by the undergraduate and graduate
students (notably by Mr Parvin Telsinghe, Mr Gaurav Kumar, Mr Chitranjan Goel, and Mr Raghavendra Rohit D.), research scholars (Dr M. Karthikeyan, Dr Jagu S. Rao, Dr Sachin Singh, Dr Mohit Lal, Dr D. J. Bordoloi, Dr C. Shravankumar, Dr Sandeep Singh, Mr Dipendra K. Roy, Mr Purushottam Gangsar, Ms Shruti J. Rapur, Mr Siva Srinivas, Ms Nilakshi Sarmah, Mr Prabhat Kumar, Mr D. Gayen and Mr Gyan Ranjan), and the project, technical, and office staff as well as the faculty at IIT Guwahati. I also thank the innumerable students and practicing engineers worldwide who approached me for clarification on their work in this field—the book makes use of some of those discussions. I would also like to show appreciation for technical support provided during production of the book by Dr. Gagandeep Singh and Ms. Mouli Sharma of CRC Press, Taylor and Francis Books India Pvt. Ltd.; Mr Glenon C. Butler, Project Editor, CRC Press and Taylor & Francis Group USA; and Ms. Christina Nyren, Production Manager & her capable team members of diacriTech, USA. This work is dedicated to my daughter, Rimjhim; son, Antariksh; and wife, Vibha, for their patience during the preparation of the book.
R. Tiwari
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Author
Dr. Rajiv Tiwari was born in 1967 at Raipur in Madhya Pradesh, India. He graduated with a BE in 1988 (Mechanical Engineering) from Government College Engineering and Technology, Raipur under Pt. Ravishankar University, Raipur, and an M.Tech. (Mechanical Engineering) in 1991 and a PhD (Mechanical Engineering) in 1997 from the Indian Institute of Technology (IIT) Kanpur, India.
He started his career as lecturer in 1996 at Regional Engineering College, Hamirpur (Himachal Pradesh), India, and worked there for one year. In 1997, he joined the Indian Institute of Technology Guwahati as assistant professor in the Department of Mechanical Engineering. He worked as a research officer at the University of Wales, Swansea, UK, for one year in 2001 on deputation. He was elevated to associate professor in 2002 and to Professor in 2007 at IIT Guwahati. He was the head of the Center of Educational Technology and Institute Coordinator of the National Programme on Technology Enhanced Learning (NPTEL) during 2005–2009, and the National Coordinator of the Quality Improvement Programme (QIP) for engineering college teachers during 2003–2009. He also visited University of Darmstadt Germany under DAAD fellowship during May-July 2011.
He has been deeply involved in various research areas of rotordynamics, especially identifying mechanical system parameters (e.g. the bearings, seals, and rotor crack dynamic parameters), diagnosing the faults of machine components (e.g. bearings, gears, pumps, and induction motors), and applying active magnetic bearings to monitoring the condition of rotating machinery. His research areas also include rolling element bearing design and analysis for high-speed applications. He has completed three projects for the Aeronautical Research & Development Board (ARDB), India on these topics. He has been offering consulting services for the last several years to Indian industries like the Indian Space Research Organisation (ISRO), Trivendrum; Combat Vehicle R&D Establishment (CVRDE), Chennai; and Tata Bearings, Kharagpur, as well as other local industries in the northeast of India. One of the European power industries, Skoda Power, Czech Republic, has also consulted him on seal dynamic parameter estimates for steam turbine applications.
Dr. Tiwari has authored more than 130 journal and conference papers. He has guided 38 M.Tech. students and 7 PhD students and 8 more are currently pursuing research projects.
He has successfully initiated and organized a national-level symposium on rotor dynamics (NSRD2003), four short-term courses on rotor dynamics (2004, 2005, 2008, 2015), and a national workshop on “Use and Deployment of Web and Video Courses for Enriching Engineering Education” (2007) at IIT Guwahati, India. He has jointly organized an “International Conference on Vibration Problems” (ICOVP 2015) at IIT Guwahati. He has developed two web- and video-based freely available on-line courses under NPTEL: (1) Mechanical Vibration and (2) Rotor Dynamics, and under MHRD sponsored virtual lab on the “Mechanical Vibration Virtual Lab.”
A Brief History of Rotor Systems and Recent Trends
This chapter presents a brief history of the rotor dynamics field. It reviews the early development of simple rotor models, starting with the Rankine to Jeffcott rotor models and physical interpretations of various kinds of instabilities in rotor-bearing systems. It also reviews the development of analysis methods for the multiple degrees of freedom and continuous systems to allow practicing engineers to apply these methods to real turbomachineries. It also summarizes work on condition monitoring and recent trends in the area of rotor dynamics.
First, however, it would be relevant to examine the importance of this subject. Also, the main difference of this subject as compared to orthodox structural dynamics will be looked into, both in terms of dynamic analysis and condition monitoring (or system identification). The rotating machinery application exists in the domestic, medical, manufacturing, automotive, marine, and aerospace fields. Predicting the dynamic behavior of such rotating machinery (Figure 1.1a) is essential to prolong the life of the machinery along with the comfort and protection of humans. Rotor dynamics covers these analyses, and hence it is imperative that rotordynamicists have a firm understanding of it. Rotor dynamics is different in the following aspects as compared to structural vibrations:
i. due to the relative motion between machine elements, rotating machineries have intrinsic forces and moments, which are often nonlinear in nature;
ii. for high-speed applications, the gyroscopic effect is very important, which provides a natural frequency as speed-dependent and the associated forward and backward whirl phenomena;
iii. similar speed dependency of natural frequency occurs due to the presence of bearings and seals;
iv. rotors can have instability due to various reasons: bearings and seals (due to cross-coupled anisotropic stiffness), asymmetric rotors (such as keyways or slots in rotors and turbine blades), internal (rotating) damping (material damping and friction between two rotating components), steam whirls (due to the high pressure of the steam in turbines), rubs (between rotor/blade/blisk and stator), and several other reasons;
v. due to high absolute motion of machine elements and due to the high relative motion among machine elements, the condition monitoring and maintenance practice also differ as compared to structures;
vi. faults in rotors can give rise to excessive dynamic loads and even lead to instability (for example, cracks in rotors);
vii. sensors and instrumentation differ due to the vibration measurement of rotating components with respect to the stator;
viii. vibration signal processing also differs due to phase measurements with respect to the rotating component reference point during balancing of rotors especially while acquiring data during
run-up and rundown of the machinery. Moreover, while observing the forward and backward whirls in the full spectrum (see Chapter 16).
These issues make the rotor dynamics subject more demanding compared to structural dynamics.
Rotating machinery can be categorized based on various factors, like the speed of operation, the power it handles, its size, and so on. Applications with varied operating speeds are 3 to 4 rpm for cement factory kilns, stone crushers, and escalators; 100 to 2500 rpm for fans; 3000 rpm for steam turbine generators; 9000 to 15,000 rpm for industrial compressors; 20,000 rpm for jet engines for airplanes; 50,000 rpm for cryogenic fuel pumps in rockets; 100,000 rpm for vacuum pumps for centrifuges; and above 100,000 rpm for micromachining applications. Similarly, these rotating machineries have varied power capacities: 0.5 to 3 W for the medical, micromachining, and household appliances; 2 to 3 MW for wind turbines and train locomotives; 10 to 190 MW for gasoline pumps, heavy machine tools, and jet engines; 600 to 1200 MW for steam turbines and fossil fuel power stations; and 2 to 10 GW for hydraulic turbines and nuclear power plants. Depending on the application, the length of the rotor could be as long as 50 m (for the multistage steam
(a)
(b)
Fluid-film bearing Discs Bush bearing mount Motor Shaft
Coupling
(c)
FIGURE 1.1 (a) A typical industrial rotor of a turbo-charger. (b) A typical rotor-bearing laboratory test rig. (c) A close view of a rotor consisting of two discs mounted on a flexible shaft.
turbine generator), 2 m for jet engines and missiles, 0.5 m for cryogenic pumps and electric motors, and few centimeters for helicopters and machine tool spindles.
A rotor is a body suspended through a set of cylindrical hinges or bearings that allow it to rotate freely about an axis fixed in space (Figures 1.1 and 1.2). Rotors can often be represented as a single beam or a series of beam elements and rigid discs. The beam is frequently considered to be flexible. Rigid discs are mounted on the flexile beam either by shrink-fit or other mechanical means. Practically, a rigid disc model represents flywheels, blades, cranks, rotary wings, coupling, disc brakes, impellers, and rolling bearings. Engineering components concerned with rotor dynamics include the rotating components of machines, especially of turbines, generators, motors, compressors, blowers, and the like. The parts of the machine that do not rotate are referred to by the general term stator. The machine element that allows relative motion of the rotor relative to the stator is called the bearing. Rotors of machines have a great deal of rotational energy and a small amount of vibrational energy while in operation. This is evident from the fact that a relatively small gas turbine propels a huge aircraft. The purpose of rotor dynamics as a subject is to keep the vibrational energy as small as possible. In operation, rotors undergo transverse (lateral or bending), longitudinal (axial), and torsional (or twisting) vibrations, either individually or in combination.
1.1 From the Rankine to Jeffcott Rotor Models
Rotor dynamics has a remarkable history of development, largely due to the interplay between its theory and its practice (Nelson, 2003). Rotor dynamics have been driven more by practice than by theory. This statement is particularly relevant to the early history of rotor dynamics. Research on industrial rotor dynamics spans more than 15 decades of history.
Rankine (1869) performed the first analysis of a spinning shaft (see Figure 1.3a). He predicted that beyond a certain spin speed “the shaft is considerably bent and whirls around in this bent form.” He defined this certain speed as the whirling speed of the shaft. In fact, it can be shown that beyond this whirling speed the radial deflection of Rankine’s model increases without limit, which is not true in actuality. However, Rankine did add the term whirling to the rotor dynamics vocabulary. Whirling refers to the movement of the center of the deflected disc (or discs) in a plane perpendicular to the bearing axis (see Figure 1.3b). In general, the frequency of whirl, ν, depends on the stiffness and damping of the rotor, as with the case of free vibration of a system (except for the synchronous whirl in which case it is equal to the unbalanced excitation force frequency, ω , i.e., the spin speed of the rotor). The whirl a mplitude is a function of the excitation force’s frequency, ω , and its magnitude. A critical speed, cr ω , occurs when the excitation frequency coincides with a (transverse) natural frequency, nf ω , of the rotor
(a) Bearing
Rotor Disc
Shaft (b) (c)
FIGURE 1.2 (a) A rigid rotor mounted on flexible bearings. (b) A flexible rotor mounted on rigid bearings.
1.3 (a) Rankine rotor model (two degrees of freedom spring-mass rotor model). (b) A Jeffcott (or Laval or Föppl) rotor model in general motion.
system and can lead to excessive vibration amplitudes. Rankine neglected the Coriolis acceleration in his analysis, which led to erroneous conclusions that confused practicing engineers for half a century.
The turbine built by Parsons in 1884 (Parsons, 1948) operated at speeds of around 18,000 rpm, which was 50 times faster than the existing engine at that time. In 1883, Swedish engineer de Laval developed a single-stage impulse steam turbine (Figure 1.3b) (named after him) for marine applications and succeeded in operating it at 42,000 rpm. He aimed at the self-centering of the disc above the critical speed (Figure 1.4), a phenomenon that he instinctively recognized. He first used a rigid rotor, but later used a flexible rotor and showed that it was possible to operate above critical speed by operating at a rotational speed about seven times the critical speed (Stodola, 1924).
In order to calculate the critical speeds of cylindrical shafts with several discs and bearings, the general theory of Reynolds (Dunkerley, 1895) was applied. The gyroscopic effect was also considered, together with its dependence on speed (i.e. a Campbell diagram—see Figure 1.5).
A Campbell diagram is designed to show rotor whirl frequency, ν, (natural frequency) with the spin speed, ω , of the shaft. In rotors the rotor whirl frequency changes with speed due to various reasons (e.g. gyroscopic effects, speed-dependent bearing parameters, etc.). However, this can be used to obtain critical speeds of rotors (intersections of whirl frequency curves with the ν = ω line, e.g., at resonance when whirl frequency is equal to the spin speed). Apart from this whirl frequency, the logarithmic decrement is also shown in the Campbell diagram, and the sign of that shows the stable/unstable stage of the rotor. Today more information is being put into these graphs, which are now are called Lee’s diagram
FIGURE
Shaft spin direction Shaft whirling direction (a)(b)
FIGURE 1.4 Synchronous whirls: (a) heavy side flying out. (b) heavy side flying in.
FIGURE 1.5 The natural frequency, υ , variation with the spin speed, ω , (a typical Campbell diagram).
(Lee, 1993). The Campbell diagram can be drawn from theoretical/numerical analyses as well as through the actual measurement from the machine.
Dunkerley found through numerous measurements the relationship known today through the work of Southwell, by which the first critical speed can be calculated, even for multidegree-of-freedom rotor cases. Dunkerley was the first to use the term critical speed for the resonance spin speed. Even with general knowledge of critical speeds, the shaft behavior at any general speed was still unclear, but more was learned from the calculation of unbalance vibrations, as given by Föppl (1895). He used an undamped model to show that an unbalanced disc would whirl synchronously with the heavy side (shown as a black spot) flying out (Figure 1.4a) when the rotation was subcritical and with the heavy side flying in (Figure 1.4b) when the rotation was supercritical. Also the behavior of Laval rotors at high speeds was confirmed by his theory. Engineers at that time were perplexed by the concepts equating Rankine’s whirling speed with Dunkerley’s critical speed. This was particularly frustrating because Rankine was far more of an authority than Dunkerley and, as a result, his predictions were widely accepted and were responsible for discouraging the development of high-speed rotors for almost 50 years. It was in England in 1916 that things came to an end. Kerr published experimental evidence that a second critical speed existed, and it was obvious to all that a second critical speed could only be attained by the safe traversal of the first critical speed.
The first recorded fundamental theory of rotor dynamics can be found in a classic paper by Jeffcott in 1919. Jeffcott confirmed Föppl’s prediction that a stable supercritical solution existed, and he extended Foppl’s analysis by including external damping (i.e. damping forces that depend upon only the absolute velocity of the rotor, whereas the internal damping comes from the rate of deformation of the shaft, often called the rotating damping) and showed that the phase of the heavy spot varies continuously as the rotation rate passes through the critical speed. We can appreciate Jeffcott’s great contributions if we recall that a flexible shaft of negligible mass with a rigid disc at the midspan is called a Jeffcott rotor (Figure 1.3b). The bearings are rigidly supported, and viscous damping acts to oppose the absolute motion of the disc. This simplified model is also called the Laval rotor and Föppl rotor, named after de Laval and Föppl, respectively.
1.2 Rotor Dynamics Phenomena Studies from Stodola to Lund
Developments made in rotor dynamics up to the beginning of the twentieth century are detailed in the masterpiece book written by Stodola (1924). Among other things, this book includes the dynamics of an elastic shaft with discs, the dynamics of continuous rotors without considering gyroscopic moment,
the secondary resonance phenomenon due to the effect of gravity, the balancing of rotors, and methods of determining the approximate values of critical speeds of rotors with variable cross-sections. He presented a graphical procedure to calculate critical speeds, which was widely used. He showed that these supercritical solutions were stabilized by Coriolis accelerations (which eventually give the gyroscopic moments). The constraint of these accelerations was the defect in Rankine’s model.
It is interesting to note that Rankine’s model is a sensible one for a rotor whose stiffness in one direction is much greater than its stiffness in the quadrature (perpendicular) direction. Indeed, it is now well known that such a rotor will have regions of divergent instability (see Figure 1.6). It is less well known that Prandtl (1918) was the first to study a Jeffcott rotor with a noncircular cross-section (i.e. elastic asymmetry in the shaft). In Jeffcott’s analytical model, the disc did not wobble or precess (see Figure 1.7). As a result, the angular velocity vector and the angular momentum vector were collinear and no gyroscopic moments were generated. This restriction was removed by Stodola.
About a decade later, the study of asymmetrical shaft systems and asymmetrical rotor systems began. The former are systems with a directional variation in the shaft stiffness (Figure 1.8a), and the latter are those with a directional difference in rotor inertia (Figure 1.8b). Two-pole generator rotors and propeller rotors are examples of such systems. As these directional differences rotate with the shaft, terms with time-varying coefficients appear in the governing equations. These systems therefore fall into the category of parametrically excited systems, in which vibrations depend on the motion itself; however, they may occur in a linear or a nonlinear system. The most characteristic property of an asymmetrical system is the appearance of unstable vibrations in some spin speed ranges. In 1933, Smith produced a pioneering work in the form of simple formulas that predicted the threshold spin speed for the supercritical instability varied with the bearing stiffness and with the ratio of external to internal viscous damping. The formula for damping was obtained independently by Crandall and Brosens (1961) some 30 years later.
Elliptical orbit due to rotor unbalance
equilibrium position
1.6 Journal center path due to perturbation: (a) stable motion. (b) unstable motion.
FIGURE 1.7 Wobbling of a disc in a rotor system. (a) A simply supported shaft with a disc near the bearing. (b) A cantilever shaft with a rigid disc at the free end.
FIGURE
Stiffness compansating slots
Magnetic pole
Slots for electrical windings y y y x x x
FIGURE 1.8 Asymmetry of the shaft and the rotor: (a) a generator rotor. (b) a three-bladed propeller.
In the early 1920s, a supercritical instability in built-up rotors was encountered. Thereafter, it was first shown by Newkirk (1924) and Kimball (1924) to be a manifestation of the rotor’s internal damping (i.e. the friction damping between rotor components). Then, Newkirk and Taylor (1925) described an instability caused by the nonlinear action of the oil wedge in a journal bearing, which was named the oil whip. Baker (1933) described self-excited vibrations due to contact between the rotor and the stator (i.e. the dry whip). The Soviet scientist Nikolai (1937) examined the stability of transverse and torsional vibrations in a shaft with a disc mounted in the center and the stability of a shaft with a disc attached to the free end. Kapitsa (1939) pointed out that a flexible shaft could become unstable due to friction conditions in its sliding bearings. In the middle of the twentieth century, Hori (1959) succeeded in explaining various fundamental characteristics of the oil whip by investigating the stability of shaft motion and considering pressure forces due to oil films. The mechanism of vibrations due to the steam whirl in turbines was explained by Thomas (1958), and that in compressors was explained by Alford (1965). The vibration of a hollow rotor containing the fluid was the problem of flow-induced vibrations. Instability due to liquids partially filling the interior cavities of rotors was demonstrated by Kollmann (1962), and in 1967 Ehrich reported that fluid trapped in engine shafts induced asynchronous vibration and changed the shape of resonance curves. Kuipers (1964) and Wolf (1968) independently succeeded in explaining the appearance of an unstable speed range in a postcritical region of a rotor system containing inviscid fluid. In the 1980s, the rotor dynamic effects of seals in fluid-handling machines received a great deal of attention. Rotor destabilization due to seals was predicted and demonstrated in an operational compressor by Jenny (1980).
As rotors became lighter and rotational speeds higher, the occurrence of nonlinear resonances such as subharmonics and superharmonics became a serious problem. Yamamoto (1955, 1957) studied various kinds of nonlinear resonances after he reported on the subharmonic resonance due to ball bearings in 1955. He also investigated combination resonances. Tondl (1965) studied nonlinear resonances due to oil films in journal bearings. Ehrich (1966) reported on subharmonic resonances observed in an aircraft gas turbine due to strong nonlinearity produced by the radial clearance of squeeze-film dampers.
The nonstationary phenomena during passage through critical speeds have been studied since Lewis (1932) reported his investigation on the Jeffcott rotor. Nonstationary phenomena that occur are one in a process with a constant acceleration (unlimited driving torque) and another with variable acceleration (limited driving torque). Natanzon (1952) studied shaft vibrations at critical speeds, and Grobov (1953, 1955) investigated in a general form the shaft vibrations resulting from varying rotational speeds. The development of the asymptotic method by Mitropolskii (1965) for nonlinear problems considerably boosted the research on this subject.
Beginning in the early 1960s, most attention focused on hydrodynamic bearings—this was largely stimulated by Lund (1964). Gunter’s work (1966) related to rotor dynamic stability problems and, combined with Ruhl and Booker’s (1972) and Lund’s (1974) methods for calculating damped critical speeds, stimulated a great deal of interest in rotor-bearing stability problems. Lund (1987) gave an overview of
(a)
(b)
the field. In the mid-1970s, rotor dynamic instability experienced with various high-pressure compressors and the high-pressure fuel turbo pump of the Space Shuttle’s main engine focused a great deal of attention on the influence of fluid–structure–interaction forces, particularly forces due to the liquid and gas seals, in pumps and turbines. Shaft seals have a similar effect as fluid-film bearings. They influence the critical speeds and can either provide damping or cause instability. Shaft seals have acquired a significant role in their effect on rotor dynamics. Someya (1989) and Tiwari et al. (2004, 2005) compiled extensive numerical and experimental results and presented literature review related to the identification of rotor dynamic parameters of bearings and seals.
Self-excited vibrations, which occur due to nonconservative forces, generally lead to large vibration amplitudes, which may ultimately damage or even destroy rotating machinery (Childs, 1993; Gasch et al., 2002; Tondl, 1965; Yamamoto and Ishida, 2001). Therefore, it is essential during the design stage of a new machine to consider the possibility of self-excitation and take measures against it. A strategy to suppress self-excited vibrations that is based on an antiresonance phenomenon (two neighboring modes having opposite effects) that can occur in parametrically excited systems (Tondl, 1978, 1991, 1998) was described by Ecker and Tondl (2004). The basic idea of parametric stabilization was adopted by introducing a time-dependent variable stiffness located at the bearing mounts. The nonconservative forces were generated through the bearings of the rotor. They showed the cancellation of the self-excited vibrations through the parametric excitation.
Shaw and Balachandran (2008) provided a comprehensive review of nonlinear dynamics of mechanical systems, including the rotating machineries. For rotating machineries they considered both the ideal and nonideal excitations. In ideal excitation case, it is assumed that the rotor speed is a specified function of time, which is a classic in the theory of nonstationary problems in dynamics and is extensively covered in the book by Mitropolskii (1965). The problem of passage through resonance of nonideal vibrating systems has received special attention from engineering researchers in recent years, but unfortunately little literature on this subject is available (Balthazar et al., 2003). Generally, nonideal vibrating systems are those for which the power supply is limited. Laval likely was the first one to work with nonideal problems via an experiment. In 1889 he built a single-stage turbine and demonstrated that in the case of rapid passage through the resonance with enough power, the maximum vibration amplitude may be reduced significantly compared with that obtained in the steady-state resonant vibration. Simultaneously, it was also known that sometimes the passage through resonance required more input power than the excitation source had available. The consequence is the so-called Sommerfeld effect in which the vibrating system cannot pass the resonance or requires an intensive interaction between the dynamic system and the motor to do it. The worst case is that of a dynamic system constructed for an overcritical operation to become stuck just before resonance conditions are reached. A strong interaction results with fluctuating motor speed and fairly large vibration amplitudes. This phenomenon was studied intensively by Sommerfeld (1927). Balthazar et al. (2003) provided an excellent review on the topic of the limited power source, in which case the system is called the nonideal vibrating system. Yamamoto and Ota (1964), Dimentberg (1961), Crandall and Yeh (1989), and Lei and Lee (1990) reported the curve veering in rotor-bearing systems. In Crandall and Yeh’s words (1989): “It is interesting that when the curve for an even rotor mode approaches the curve for an even stator mode, or an odd rotor mode approaches an odd stator mode, the two modes form a coupled system and the curves repel each other avoiding an intersection.” This was in reference to the Campbell diagram of the natural frequency of a uniform rotor rotating in a uniform stator, as a function of rotational speed.
Instability from fluid-film bearings and shaft seals arises from the fact that, during the radial displacement of a rotor, a restoring force is produced that has a component at right angles to this displacement. The phenomenon of instability was described in detail by Newkirk (1924), whose interest was in turbomachineries. The cause of this instability, in fact, lay in the oil-film bearings. In the following years it was established that in a few cases, internal friction or damping could indeed be a cause of instability. The designer must thus be aware of these possibilities. Some of the important phenomena in rotorbearing systems, its main causes, and investigators’ details are summarized in Table 1.1.
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«Ne poljubljaj, ne objemlji, Ne govori, da me ljubiš ...
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Od sramote, od kesanja.»
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Ko sem vstopil, je dvignila glavo, pogledala me boječe s svojimi velikimi, temnoobrobljenimi očmí ter se prisiljeno nasmehnila. Potem pa se je obrnila v steno.
Šèl sem za okroglo mizico pred divanom. Po nji so ležala raztresena rožnata, parfumirana pisma poleg drobnih kuvert. Nekateri listi so bili razstrti, kakor šele pol prečitani, krog druzih pa se je vil razvezan omot bledosinje barve.
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«Lahka in svobodna mi je duša, kadar sanjam o Tebi, — kakor škrjančeva pesem o jutranji zori. Vzdramil sem se iz težke noči in jasna, vesela gorkota mi je zasijala v obraz. Kakó sem se otresel pozemskega prahú, kakó sem vrgel raz sebe zarjavele okove! ...
Pred mano ni ničesar in ne za menój, v tej čisti ljubezni leží neizmerna večnost ... Ali je ne čutiš, ljubica? Ali ne vidijo bogá Tvoje nedolžne očí, kakor ga vidim jaz? ...»
In moja ljubica je ležala na divanu in gledala v steno ...
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Postajal sem nervozen in nezadovoljen. In kakor sem se jih branil, prihajali so tisti dnevi z vso silo prédme, bližali se mi od vseh stranij, kakor bi se me dotikale po udih mrzle, mokre róke. Nisem jih maral gledati, a postajali so vsak trenotek jásnejši in živéjši. Rogale so se mi moje tedanje besede z vsemi akcenti in vzdihi vred; slišal sem jih takó razločno, kakor bi jih nehoté ponavljal sam. In videl sem njene velike, vprašujoče oči, polne sreče in ljubezni; slišal sem njen otroški smeh, kakor bi padali kristali na srebrn krožnik ...
Dà, — «velik je najin svet in krasen ...» Dvoje pierrotov na pepelnično jutro ... Preveč je bilo šampanjca! ... Od telesa visé samó še raztrgane pisane cunje; lica so zamolkla in v prsih je prazno ...
Dvignila je glavo, da so se ji vsuli lasje raz rámen čez prsa. Zdelo se mi je, da vidim nocoj prvikrat njen rumenkastobledi, od strasti in življenja upali obraz, njene globoke, meglene oči, polne sramú in utrujenosti, — da čutim prvikrat njeno zoperno mehko, opolzko teló, njene mokre, težke lase, — da slišim njen tihi, boječi, odživeli glas in njene poludušene vzdihe ...
Odšel sem tiho po mehki preprogi, v zrcalu pa sem videl, da je ležala na divanu mirno, kakor ob mojem prihodu, s polzatisnjenimi očmi in povešeno golo roko.
In dvíga in širi se vroče obzorje, Na nebu razpaljenem solnce gorí, Vsa zemlja potaplja se v démantno morje, V polspanju, utrujena v prahu leží. Tam daleč ječí preperelo drevje, Kot gole, koščene roké
Razpénja in krči temno vejevje.
V očí skelí zrak težák in suh, Na prsa pada opojni duh
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In v dušo kipečo in v srca dnò
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In v prahu nesvesten klečí pred teboj
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Noč brezupna, — in nikdár več
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Glej, vse moje rosne, duhteče rože, Nocoj so razpale in v prahu ležíjo, In solnce, to lepo, velíko solnce
Ne vrne se níkdar in níkdar več ...
Najtíšjega zvoka drhteče strune, Ne bledega žarka oddaljene zvezde —
In ti si pred mano ... jaz čutim tvoj pógled ...
Ah, nagni se k meni, objemi me! —
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Na tvoje ljubeče, mrzle róke —
Ah, nagni se k meni, ti zaželjena,
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ROMANCE.
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V razkošnih, mehkih valih.
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V tem burnem plesu od strastí
Telesa trepetajo,
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Iz zlatih harp takó sladkó
Vro pesmi koprneče
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Ob belo róko Salomon
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Brezupno k njemu dvigajo Koščene, vele róke, Umirajoč strmíjo vanj Očí temné, globoke.
In zlate strune pojejo Mrtvaško melodijo, —
Poslednji vzdihi sužnjih prs Nocoj iz njih zveníjo ...
Ob belo roko Salomon Opira trudno glavo, Na čelu mu leží oblak, Okó strmí v daljavo ...
In láhno, láhno k njemu se Gorkó teló privije, Po vsem životu čut mehák Sladkó se mu razlíje.
«Zakaj ti je okó temnó
In duša nevesela?
Zakaj ti bleda žalost je Nocoj srce objela?
Ljubezen tvoja že mrjè, Še komaj porojena, —
Kaj hočem jaz brez njé, povej, Nesrečna, zapuščena?» ...
«Nelepe sanje, ljubica, Poslal mi je Jehova, — Zakaj, to sam on vé, — modrost Velika je njegova ...
Ah, nagni k meni, Sulamit, Prebele prsi svoje, — Pogíne naj ves Izrael, — Ti si kraljestvo moje!»
Ob grobu tiranovem.
Upognil zdaj si glavo, car, Če nisi je doslej nikdar.
Ves svet je klečal pred teboj, Ves svet široki bil je tvoj.
Oj car, oj car, pa zapovej Naj bela smrt beží naprej!
Že krsta iztesana je, Že postelj ti postlana je ...
Oj to vam je vesel pokop, Ves svet hití pred carjev grob.
A ko ga položé v zemljó Okó nobeno ni mokró.
Okó nobeno ni mokró, Srcé nobeno ni težkó.
Med ljudstvom patrijarh stojí. Veselo vzdihne, govorí:
«Oj hvala, hvala ti, Gospod, Da osvobodil svoj si rod!
Dovolj je v solzah vzdihoval, Trpljenja pač dovolj prestal!
Jaz begal sem okrog pregnan, Pregnan, povsod zaničevan.
In ti poslal si belo smrt, Tiran mogočni v prah je strt.
Spet dal si meni svetlo čast, Spet dal si meni vso oblast.
Oj hvala, hvala ti, Gospod, Da osvobodil svoj si rod!»
Med ljudstvom knez visok stojí
In patrijarhu govorí:
«Oj patrijarh, nikar, nikar, Vladánje pač ni tvoja stvar!
Dovolj je roki tvoji križ, Zakaj po žezlu hrepeniš? ...
O čuvaj sívo si glavó, Na pot ne hodi mi drznó!»
Ob knezovih besedah teh Vsem divja strast vsplamtí v očeh.
Še tristo knezov tam stojí, Po žezlu vsakdo hrepení.
Po žezlu vsakdo hrepení, Že v roki vsakdo meč drží.
Ne nagne še se beli dan, Pa zadivjá že boj strašan ...
Veselo, kakor še nikdar, Smehljá se v krsti — mrtvi car.
Ivan Kacijanar.
Kacijanar, vojskovodja slavni,
Ves zamíšljen hodi po šatoru.
Težka skrb mu polni trudno glavo, Ríše mu na čelo temne črte
In obrvi mu ježí košate.
Polumesec na šatorih belih
Ob ostrogu blíska se krščanskem ...
Ne rešítve in ne krasne zmage, —
Sam in truden, — kakor ranjen sokol
Brez pomóči v jati lačnih vranov ...
In globočja bol v junaških prsih, In bridkejša skrb na čelu resnem.
«Kaj pomeni tam ta šum skrivnostni?
Saj prišlò še ní krvavo jutro.»
In služabnik stari, zvesti Miloš,
Odgovarja: «Gospodar, ne skrbi!
Res prišlò še ni krvavo jutro, Ali srca jim kipé junaška,
Sèn jim lahki na očí ne more ...»
Zopet praša slavni Kacijanar:
«Kaj pomeni tam ta šum skrivnostni, Kot hodile tolpe bi po planem?
Saj še vstalo ní krvavo solnce.»
Odgovarja mu s tresočim glasom:
«Oj ne skrbi, gospodar mogočni!
Res še vstalo solnce ní krvavo, Vso ravnino krije noč ledena; A v telesu mrzlem — mrzla duša, In junaki hodijo po planem, Da si mrzle ude oživíjo ...»
Tretjič praša slavni Kacijanar:
«Kaj pomeni ta topòt skrivnostni, Kot bi v dalji dírjali konjiki?
A govori mi resnico, Miloš!
Priča mi je tvoj pogled brezupni, Da zakrívaš mi srcé nemirno.»
In zastoče grenko stari Miloš,
Odgovarja solzen gospodarju:
«Kaj mi né bi se okó solzílo, Kaj mi né bi srce trepetalo!
Gledal tvojo sem visoko slavo;
Tvoja videl sem junaška dela —
Oj število krasno ... Vse za tabo;
Izgubljen si, Kacijanar silni! ...
Kaj pomeni tam ta šum skrivnostni?
Vojska naša je v sramotnem begu.»
Ne vztrepeče Kacijanar slavni, Ne porósi mu okó se bistro; Meč opaše, pa zasede konja
In odjaše v diru skozi meglo; Poleg njega jaše stari Miloš.
Hladna megla krije vso ravnino; In ne gane se peró na drevji, In ne gane bilka se na polju;
Vse je tiho kakor pod gomilo ...
«Ali vidiš tam bleskèt orožja, Ali vidiš prápore razvíte?»
«Kacijanar, ózri se v daljavo, —
Žarki zore se bleščé po hribih, Nad gozdovi plava bela megla ...»
